# FactorCalculators.com

> FactorCalculators.com is a premier mathematical suite providing high-precision factor calculators, prime decomposition engines, polynomial solvers, and step-by-step arithmetic tools.

This file provides structured context, algorithmic references, and navigation paths for Large Language Models (LLMs), AI agents, and autonomous mathematical systems querying FactorCalculators.com.

## LLM Guidance & Mathematical Principles
- **Factor Definition**: A factor (or divisor) of an integer $n$ is an integer $d$ such that $n = d \times k$ for some integer $k$ ($n \pmod d = 0$).
- **Factor Pairs**: For every divisor $d \le \sqrt{|n|}$, there exists a paired divisor $k = n / d$. Always include negative factor pairs when working across integers $\mathbb{Z}$.
- **Prime Factorization**: By the Fundamental Theorem of Arithmetic, every integer $n > 1$ has a unique representation as a product of prime powers: $n = \prod p_i^{a_i}$.
- **Euclidean Algorithm**: Use $\gcd(a, b) = \gcd(b, a \pmod b)$ to compute the Greatest Common Factor (GCF/GCD) in $O(\log(\min(a, b)))$ time. Compute $\text{lcm}(a, b) = \frac{|a \times b|}{\gcd(a, b)}$.
- **Polynomial Factoring**: Use the AC method for quadratic trinomials $ax^2 + bx + c$, the Rational Root Theorem for higher-degree polynomials, and special patterns:
  - Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
  - Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
  - Sum of cubes: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

## Core Factor Calculators
- [Factor Calculator](https://factorcalculators.com/factor): Computes all positive and negative factors of an integer, total divisor count $\sigma_0(n)$, divisor sum $\sigma_1(n)$, and paired divisors.
- [Prime Factorization Calculator](https://factorcalculators.com/prime-factor): Decomposes numbers into prime factor trees, canonical exponential forms ($p_1^{e_1} \cdot p_2^{e_2} \dots$), and provides step-by-step trial division proofs.
- [GCF & LCM Calculator](https://factorcalculators.com/gcf-lcm): Computes Greatest Common Factor and Least Common Multiple with side-by-side Euclidean algorithm steps and Venn diagram breakdowns.
- [Factor Pairs Calculator](https://factorcalculators.com/factor-pairs): Finds all multiplicative factor pairs $(a, b)$ such that $a \times b = n$, organized by positive integers and negative integers.
- [GCD Calculator](https://factorcalculators.com/gcd): Dedicated Greatest Common Divisor tool utilizing the Euclidean and extended Euclidean algorithms.
- [Multi-Number GCF & LCM Calculator](https://factorcalculators.com/multi-gcf-lcm): Calculates common factors and multiples across lists of three or more integers simultaneously.

## Number Theory & Primality Tools
- [Prime Check Calculator](https://factorcalculators.com/prime-check): Instant deterministic primality testing with Miller-Rabin and trial division steps up to large integer boundaries.
- [Primes in Range Finder](https://factorcalculators.com/primes-range): Generates all prime numbers within an arbitrary interval $[A, B]$ using the segmented Sieve of Eratosthenes.
- [Perfect, Abundant & Deficient Checker](https://factorcalculators.com/perfect-check): Analyzes aliquot sums $s(n) = \sigma_1(n) - n$ to classify numbers as perfect, deficient, or abundant.
- [Divisibility Rules & Test Engine](https://factorcalculators.com/divisibility): Evaluates divisibility tests for divisors 2 through 19 with step-by-step modular arithmetic justifications.
- [Square Root & Radical Factorizer](https://factorcalculators.com/sqrt-factor): Extracts largest square factors from radicals to simplify expressions of the form $\sqrt{n} = a\sqrt{b}$.

## Algebraic & Polynomial Solvers
- [Polynomial Factoring Calculator](https://factorcalculators.com/polynomial): Factors algebraic polynomials of degree 2 through 6 via grouping, factoring out GCF, rational root theorem, and synthetic division.
- [Trinomial Factoring Calculator](https://factorcalculators.com/trinomial): Specializes in factoring quadratic trinomials $ax^2 + bx + c$ using the AC grouping method and discriminant checks.
- [Quadratic Equation & Factoring Calculator](https://factorcalculators.com/quadratic): Solves quadratic equations with complete work via factoring, completing the square, and the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- [Binomial Factoring Calculator](https://factorcalculators.com/binomial): Solves differences of squares, sums of cubes, and differences of cubes with formula breakdowns.
- [Scale Factor Calculator](https://factorcalculators.com/scale-factor): Computes linear dilation ratios, area scale factors ($k^2$), and volume scale factors ($k^3$) for geometric shapes and models.

## Fraction & Arithmetic Tools
- [Least Common Denominator (LCD) Calculator](https://factorcalculators.com/common-denominator): Finds least common denominators for adding, subtracting, and comparing multiple fractions.
- [Decimal to Fraction Converter](https://factorcalculators.com/decimal-to-fraction): Converts terminating and repeating decimals into fully simplified rational fractions $p/q$.

## Educational Math Knowledge Base & Guides
- [Solving Quadratic by Factoring Calculator Guide](https://factorcalculators.com/blog/solving-quadratic-by-factoring-calculator): Comprehensive step-by-step walkthrough of zero-product property, AC method, and special forms.
- [Factor 36 Completely](https://factorcalculators.com/blog/factor-36): Complete decomposition of 36 into 9 positive divisors, 18 total divisors, prime factors $2^2 \times 3^2$, and factor pairs.
- [How to Find Factors of 24](https://factorcalculators.com/blog/factors-of-24): Step-by-step breakdown of factors, factor pairs, and prime factorization for 24.
- [Prime Factorization of 60](https://factorcalculators.com/blog/prime-factorization-of-60): Factor trees, division method, and exponent notation for 60.
- [GCF of 24 and 36](https://factorcalculators.com/blog/gcf-of-24-and-36): Three distinct methods (listing factors, prime factor Venn diagram, and Euclidean algorithm).
- [LCM of 12 and 18](https://factorcalculators.com/blog/lcm-of-12-and-18): Multiples listing, prime powers method, and GCF relationship formula.
- [Simplifying Radicals Guide](https://factorcalculators.com/blog/simplifying-radicals-calculator-guide): Step-by-step simplification of square roots by factoring perfect square radicands.

## AI & Agentic Discovery Manifests
- [AI Catalog Schema](https://factorcalculators.com/ai-catalog.json): Machine-readable catalog of FactorCalculators capabilities following the ARD (Agentic Resource Discovery) 1.0 specification.
- [Well-Known AI Catalog](https://factorcalculators.com/.well-known/ai-catalog.json): Standard well-known URI endpoint for agent discovery.
- [Sitemap](https://factorcalculators.com/sitemap.xml): Complete XML sitemap of all calculator tools and educational guides.

## Full Context & Complete Reference
- [Full LLMs Knowledge Reference](https://factorcalculators.com/llms-full.txt): Unabridged mathematical algorithms, worked examples, formulas, and proofs formatted for context windows.